Download Slides for lecture #23

Lecture 23: Polycrystalline Materials
PHYS 430/603 material
Laszlo Takacs
UMBC Department of Physics
Continuity has to be maintained by
• slip and twinning in individual grains,
• geometrically necessary dislocations,
• sliding along grain boundaries, grain rotation.
Grain structure before & after plastic deformation
e
e
Pl
Pl
Differently oriented grains slip along different directions, try to deform
and rotate differently. The grain structure changes and surface
roughness develops - a problem in rolling metals into sheets.
From the Ph.D. Thesis of Eric Moore, 2006
Sample annealed 425°C for 1 h
Grain structure and the orientation change of
individual grains during deformation
[1 1 1]
[1 1 1]
Tensile
Direction
[1 1 1]
Tensile
Direction
Tensile
Direction
Grain 5
Grain 1
Grain 4
[0 0 1]
[1 0 1]
[0 0 1]
[1 0 1]
e=0
4
e=0
6
3
3
Tensile
Direction
3
[1 1 1]
Tensile
Direction
Grain 3
Grain 6
Grain 2
[0 0 1]
[1 0 1]
[0 0 1]
[1 0 1]
From the Ph.D. Thesis of Eric Moore, 2006
[0 0 1]
6
~100 mm
10o
[1 1 1]
[1 1 1]
4
2
e = 0.140
~100 mm
Tensile
Direction
5
1
4
2
6
e =0.140
5
1
2
[1 0 1]
Sample
Normal
e =0.140
5
1
[0 0 1]
[1 0 1]
Dislocation pile-up
Dislocations cannot cross grain
boundaries (except for rare cases
involving coherent boundaries.)
There are several dislocations moving
toward the grain boundary in a slip
plane. They repel each other with a
force
Gb2 1
F
2 (1  ) r
A pile-up forms with increasing
distance between dislocations with
increasing distance from the grain
boundary. The first dislocation is
pushed toward the grain boundary by
the effect of the external stress plus
the force from all the other
dislocations. This large stress is
transmitted into the neighboring grain
also.
Typically strength increases with decreasing grain size - an important benefit that
drives the development of ultrafine grained materials. These are typical stress - strain
curves for Cu to 22 nm grain size. How can this be explained by dislocation pile-up?
From Khan, Farrokh, Takacs, J. Mater. Sci. 43 (2008) 3305
The Hall-Petch relation
The number of dislocations that can fit into
a grain of diameter D is limited to
 (1  ) D
n

Gb 2
At the tip of the pile-up, the external stress
plus (n-1) dislocations act, each with a
force
 b. The local stress at the grain
boundary is n. This stress penetrates
neighboring grain 2, where at x0
 2 (x 0 )  m2   (x 0 )

Data available for Cu as a function of grain
size. Notice that hardness begins to decline
for very small grains. There is an optimum
where ultimate shear stress, tensile stress,
and hardness are the largest.
 (1  )
2Gb
D 2
First term is from external stress with the
Schmid factor of grain 2, second term is
stress from grain 1,  describes geometry.
At the yield point 2 reaches 0, the
ultimate shear stress. If nothing but the
grain size changes and the first term is
negligible
D  2  const.  k y
For a single crystal the ultimate
stress is 0, Adding the term from grain 2
2

  0 
k y
D
•
•
•
For grain size larger than 100 nm - 1µm depending on the material, H-P applies.
Deviations become increasingly significant, until a maximum is reached at about 20 nm
(less than 500,000 atoms).
Softening takes place for smaller grain sizes. The deformation mechanism relies more
on grain boundary processes than dislocations. This is an area of current research.
How is it possible to make materials with such small grain size?
• Inert gas condensation, solution processing.
• Rapid solidification, crystallization of an amorphous initial state.
• Sever plastic deformation, ball milling.
Grain size refinement is an important strengthening mechanism. Are there others?
Dislocations move relatively freely in defect-free single crystals. Any
deviation - solute, other dislocation, grain boundary, etc. - acts as an
obstacle, impeding dislocation motion and thus increasing strength.
Notice that the large increase of critical shear stress in Ag-Au is in spite of
the chemical and structural similarity of Au and Ag.
Solid solution hardening
• Parelastic interaction - lattice parameter effect
Solute - both larger and smaller than the matrix - can find an energetically
favorable place in a dislocation. The energy difference of the solute between a
general lattice site and a site in the core of a dislocation becomes a barrier to
dislocation motion. Affects mostly edge dislocations, although screw can be
affected if impurity causes non-isotropic distortion (such as C in bcc Fe.)
• Dielastic interaction - shear modulus effect.
Dislocation energy is proportional to the shear modulus. If an impurity changes
G, it changes the dislocation energy, providing a barrier to dislocation motion.
• Suzuki interaction - chemical effect.
Impurity changes the stacking fault energy, thus the separation between partial
dislocations. Dislocation motion requires restoring the original separation.
• The magnitude of solid solution hardening (from parelastic an dielastic
effect) is proportional to the square root of the concentration:
 c  G      2 c a /3
3
where  

d ln a
d ln G
,


,   const. 1
a
a
dc
dc
Solid solution hardening
of Cu by several solutes.
Notice that the classical
bronze-forming elements
Sn and As provide much
hardening. The good
properties of brass (Zn)
are due to the formation
of compound phases, not
due to solid solution
hardening.
The effect of a metal as
alloying element is very
different from its own
mechanical properties.
Usually we assume that the dislocations
move but the motion of the solutes is
negligible.
If the solutes diffuse fast - e.g. C in iron time-dependent phenomena result, such as
the Lüders maximum.
Once the dislocation breaks free from the
impurities, it moves easier and the stress
decreases. But if the test is stopped for some
time, the impurities diffuse into the
dislocations again and the dislocation must
break free again.
Dispersion hardening
Introducing finely dispersed particles into a
metal is often a difficult task, as the
particles must be wetted by the matrix but
not dissolved by it. Searching for a way to
disperse aluminum oxide particles in Nibased superalloys lead to the development
of mechanical alloying in the late 1960s.
Finely dispersed particles are
obstacles for dislocation motion.
Orowan mechanism: If the
dislocation meets a pair of
particles, it can only proceed by
bowing out, similar to the
principle of the Frank-Reed
source. The stress needed to
move the dislocation across this
barrier is
 = Gb/(l-2r)
Precipitation hardening
a form of dispersion hardening
Precipitates form when a solid
solution becomes supersaturated
during cooling and a second phase
crystallizes in the matrix in the form
of small crystallites. They are often
coherent (like Ni3Al in Ni.)
Precipitates are often coherent or semicoherent. A dislocation can pass through a
coherent precipitate, but it requires extra
stress as a stacking fault results and
creating it requires energy. This mode
dominates, if the material contains many
small particles. If the same total volume is
in fewer but large particles, the Orowan
mechanism is preferred. The largest
hardness is achieved when the two
mechanisms require the same stress; this
happens at around 20 nm.
Strain rate dependence
The time / strain rate dependence of the
stress-strain curve is intuitively anticipated
and clearly observed, but it is very difficult to
explain quantitatively. Typically it is
characterized by the strain rate sensitivity,
m, defined as
m
Ball milled and consolidated Cu,
average particle size 32 nm.
Babak Farrokh, UMBC

d ln 
Ý
d ln e
Temperature dependence
•
•
•
•
The strain rate sensitivity is low at low temperature (T < 0.5 Tm) but
increases at higher temperature. This is understandable, as atomic motion
is more vigorous at higher temperature, diffusion is faster.
High strain rate sensitivity is usually associated with larger strain to failure.
Consider a tensile experiment. If a random cross section decreases in
diameter, the strain rate at that cross section increases, with enough strain
rate sensitivity the section becomes harder and no further reduction leading
to failure occurs.
In fine grained (<10 µm) materials close to Tm very large strain rate
sensitivity and strain to failure (up to 100-fold elongation) can be observed.
This is called superplasticity. It depends on grain boundary sliding, rather
than dislocation mechanisms.
Nanocrystalline materials contain many grain boundaries, superplasticity
should be more easily achieved.
Superplasticity of electrodeposited nc Ni and
nc Al-1420 alloy and Ni3Al by severe plastic deformation
Notice that superplasticity was achieved at a
temperature much below typical for conventional
materials; 350°C for Ni corresponds to 0.36 Tm!
McFadden et al. (UC Davis, Ufa, Russia)
Nature 398 (1999) 684-686
High RT ductility of a hcp Mg-5%Al-5%Nd alloy
Ball milling results in repeated
fracturing and agglomeration of
grains, resulting in a nanometer
scale microstructure (mean grain
size probably 25 nm). Grain rotation
and sliding results in high ductility
even at room temperature and
3x10-4 s-1 stress rate. (Recall that a
hcp material is normally brittle.)
L. Lu and M.O. Lai, Singapore
Ceramic nanocomposite of 40 vol.% ZrO2, 30% Al2MgO4, and 30% Al2O3
shows superplastic behavior at 1650°C.
Kim et al. (Tsukuba, Japan)
Creep
mass flux
Nabarro-Herring creep
volume diffusion
Coble creep
mediated by
grain boundary diffusion
Anelasticity and viscoelasticity
•
•
•
Small time dependent effects can be observed also for elastic
deformation - e.g. related to reversible diffusion of C in a steel
under stress.
If a sample is vibrated close to resonance, the deviation from
perfect elasticity, i.e. the existence of dissipative processes,
results in a change of the resonance curve.
While technologically unimportant, this is the way one can gain
information about diffusion and other time dependent
phenomena at low temperature, where their rate is very low and
the macroscopic effects are not detectable.